QEtaKolberg¶

cite[Def.~35]{Radu:RamanujanKolberg:2015}

chiExponent: (Integer , PositiveInteger , List NonNegativeInteger ) -> Integer: chiExponent(sigmaInfty, m, orb) computes e such that $exp(2pi i e / 24) = chi_{s,m,t)$. We assume $sigmaInfty = sum_{delta|mm} delta s_delta$. See Lemma ref{thm:chi-exponent}. Furthermore, we assume that orb = orbit(sigmaInfty, m, t).

etaCoFactorSpace: (PositiveInteger , PositiveInteger , List Integer , PositiveInteger , NonNegativeInteger ) -> Record(particular: Union(Vector Integer , failed), basis: List Vector Integer ): etaCoFactorSpace(nn, mm, s, m, t) returns a vector v and the basis of a space such that #v=#divisors(nn) and modularConditions?(nn, members r, mm, s, m, t) is true for any r = v + reduce(_+, [z.i * basis.i for i in 1..#basis]) and any sufficiently long list z of integers. The function fails, if there is no such solution.

modularConditions?: (PositiveInteger , List Integer ) -> Boolean: modularConditions?(nn, r) returns true iff the eta quotient corresponding to r is a modular function for Gamma_0(nn). It is equivalent to modularConditions?(nn, r, 1, [0], 1, 0). Compare with rStarConditions from QAuxiliaryModularEtaQuotientPackage.

modularConditions?: (PositiveInteger , List Integer , PositiveInteger , List Integer , PositiveInteger , NonNegativeInteger ) -> Boolean: modularConditions?(nn, r, mm, s, m, t) returns true iff all the conditions of Theorem~ref{thm:RaduConditions} are fulfilled. Compare with rStarConditions from QAuxiliaryModularEtaQuotientPackage.

modularInputConditions?: (PositiveInteger , PositiveInteger , List Integer , PositiveInteger , NonNegativeInteger ) -> Boolean: modularInputConditions?(nn, mm, s, m, t) returns true iff all the conditions for the parameters are fulfilled. This checks whether (nn, mm, s, m, t) is in Delta^* as defined in Definition~ref{def:modular-input-conditions}.

orbit: (Integer , PositiveInteger , NonNegativeInteger ) -> List NonNegativeInteger: orbit(sigmaInfty, m, t) computes the elements of $modularOrbit{s,m,t)$ as defined in qetafun.spad, cite[Def.~42]{Radu:RamanujanKolberg:2015} and cite[Lemma 4.35]{Radu:PhD:2010}. We assume $sigmaInfty = sum_{delta|M} delta s_delta$.

orbitRadu: (Integer , PositiveInteger , NonNegativeInteger ) -> List NonNegativeInteger: Do not use this function! It is just another implementation of orbit.